In real analysis (and other applications like the graph-theoretic structural approach to sparse matrices), it would sometimes be convenient to quantify over all generic elements. A generic element $\xi$ would basically satisfy $(\exists xf(x)\neq g(x))\to f(\xi)\neq g(\xi)$ for all functions $f$, $g$ relevant to the current context. So a generic element quantifier would be a kind of "last word" quantifier, i.e. it has to wait until all other context is laid down, before it can fix its meaning. But even as a "last word" quantifier, it seems to be more ambiguous and brittle then other "overly expressive" quantifiers like second order quantifiers or Henkin quantifiers.
Let's assume that the first order language (with equality) contains only function symbols, but no predicate symbols. Is it possible to give a semantics for generic element quantifiers in this case? If this shouldn't be possible in general, even in the weak sense of full semantics for second order quantifiers, then maybe it helps if we restrict the allowed models to $\mathbb R^n$?
If it is possible to give a semantics for generic element quantifiers, do they extend the expressive power of the corresponding logic significantly?